Sure! Let's explore the transformations of the function [tex]\( f(x) = -\sqrt{2x} + 7 \)[/tex] based on the parent function [tex]\( f(x) = \sqrt{x} \)[/tex].
1. Reflection:
- Reflection over the x-axis: The negative sign in front of the square root, [tex]\(-\sqrt{}\)[/tex], indicates that the graph is reflected over the x-axis. This means that every point on the parent function [tex]\( f(x) = \sqrt{x} \)[/tex] is flipped to the opposite side of the x-axis.
2. Dilation:
- Horizontal compression by a factor of 1/2: The coefficient [tex]\( 2 \)[/tex] inside the square root, [tex]\(\sqrt{2x}\)[/tex], indicates a horizontal compression. Specifically, the factor inside the square root (2) results in a compression by a factor of [tex]\(1/2\)[/tex]. This means that the function will grow at half the rate when compared to the parent function.
3. Vertical Shift:
- Upward by 7 units: The constant [tex]\( +7 \)[/tex] at the end of the function indicates a vertical shift upward by 7 units. This means that each point on the graph of the reflected, horizontally compressed function will be moved up by 7 units.
Here's how you'd complete the dropdowns:
- Reflection: Over the x-axis
- Dilation: Horizontal compression by a factor of 1/2
- Vertical Shift: Upward by 7 units
There is no horizontal shift in this function, so that dropdown should remain empty or indicate no horizontal shift.
These steps outline all the transformations of the function [tex]\( f(x) = -\sqrt{2x} + 7 \)[/tex] from the parent function [tex]\( f(x) = \sqrt{x} \)[/tex].
